191k views
3 votes
One of U1 and U2 is a subspace of V while the other is not. Determine which is which and explain the criteria for subspace inclusion.

1 Answer

4 votes

Final answer:

To determine if U1 or U2 is a subspace of V, we need to check for closure under addition, scalar multiplication, and containing the zero vector.

Step-by-step explanation:

In order to determine whether U1 or U2 is a subspace of V, we need to check if they satisfy three criteria: closure under addition, closure under scalar multiplication, and containing the zero vector.

To check closure under addition, we need to verify if for any two vectors in U1/U2, their sum is also in U1/U2.

To check closure under scalar multiplication, we need to verify if for any scalar c and any vector in U1/U2, the product of c and the vector is also in U1/U2.

Finally, to check if U1/U2 contains the zero vector, we need to ensure that the zero vector is in U1/U2.

If U1 satisfies all three criteria, then U1 is a subspace of V. Otherwise, U2 is the subspace of V.

User Nantucket
by
8.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories